Homeschooling High School Math

Feature photo (above) by Sphinx The Geek via flickr.

Most homeschoolers feel at least a small tinge of panic as their students approach high school. “What have we gotten ourselves into?” we wonder. “Can we really do this?” Here are a few tips to make the transition easier.

Before you move forward, it may help to take a look back. How has homeschooling worked for you and your children so far?

If your students hate math, they probably never got a good taste of the “Aha!” factor, that Eureka! thrill of solving a challenging puzzle. The early teen years may be your last chance to convince them that math can be fun, so consider putting aside your textbooks for a few months to:

• Subscribe to Games magazine.
• Read Brian Bolt books, or work through Raymond Smullyan’s What is the Name of This Book?
• Design your own tessellation T-shirts for Christmas gifts.
• Remodel the house. From financing to floor coverings, that is real math in action.

On the other hand, if you have delayed formal arithmetic, using your children’s elementary years to explore a wide variety of mathematical adventures, now is a good time to take stock of what these experiences have taught your students.

• How much of what society considers “the basics” have your children picked up along the way?
• Are there any gaps in their understanding of arithmetic, any concepts you want to add to their mental tool box?

Transition Math

Many homeschoolers take advantage of the free videos and computer-graded quizzes at Khan Academy to review and consolidate their students’ skills before tackling high school math. My daughter enjoyed Khan Academy for a couple of weeks, but then the mindlessly-repetitive nature of their randomly-generated problems began to irritate her.

I’d rather use a textbook such as Basic College Mathematics (choose from several versions by various authors) that has old editions available for pennies — there’s no need to pay new-book prices, since arithmetic hasn’t changed in centuries. A textbook offers better problems, carefully selected and designed to build on each other. You can use the chapter review problems as a pre-test and study only the topics your child needs. (Buddy Math is great for high school!)

But beware: many books and online resources present mathematics primarily as a follow-the-rules subject. That is fine for review, but if you find a concept that is new to your student, don’t be satisfied with just teaching the procedure. Memorization can give your students quick “success,” producing page after page of correct answers, but for most people, memorized rules will not stick in the mind well enough to support future learning.

If you don’t understand a math topic yourself, dig for more information about it. Ask why it is true and how it relates to other topics. Find out how it connects to the deep ideas of mathematics, such as symmetry, functions, change, etc.

One excellent way to explore the meaning of mathematics is through Herb Gross’s free online Math as a Second Language courses.

You and your student may need to search out several explanations and work your way through dozens of sample problems in order to wrap your minds around a new topic. Wrestling a math concept into submission can be hard! But the work is worth it, because when you are able to help your student truly understand what the math is doing, that concept becomes a rock-solid foundation on which to build.

The Meat of High School Math

If your children plan to go to college, they will benefit from a serious amount of mathematics in high school. Even those who don’t plan on college should learn as much as they can because we live in a technological society, and technology runs on math.

As mathematician and author Oleg Gleizer says, “Math is freedom. If we don’t know math, our choices are so limited.”

This does not mean that every student must follow the traditional algebra-to-calculus approach of school mathematics. Creative students who want to blaze their own trail of interest-led learning can master the basics of mathematics and then branch out to explore many non-traditional topics, such as logic, combinatorics, or game theory.

But first, what are the basics of high school mathematics? What math does every student need in order to be a math-literate member of our technology-based society?

Everyone Needs Algebra

Many people think that algebra is “the rules and procedures for doing math with letters,” and they can’t imagine ever using it in real life. But algebra is much deeper, richer, and more important than just rules and procedures.

Algebra is:

• Using the things you know to figure out something you don’t know.
• Thinking about the relationships between things, how changing one thing affects everything else.

Things we know, things we don’t know, and the relationships between them — such is the stuff of real life. Algebra gives us the tools to model, understand, and solve the problems we face at home and at work, from shipping to science, politics to planning retirement.

Everyone Needs Geometry

Geometry is not just formulas, two-column proofs, and memorizing stuff to pass a test. Like algebra, geometry is deeper, richer, and more important than most people realize.

Geometry is:

• Using the shapes and properties we know to figure out something we don’t know.
• Thinking about the relationships between things, how changing one thing affects everything else.

We live in a three-dimensional world, surrounded by length and height and breadth. From crafts to construction, marketing to medicine, property development to playing in virtual worlds — shapes and volumes and the relationships between them affect almost everything we do.

Everyone Needs Statistics

No one can be considered an educated adult in our society without a basic understanding of statistics, and especially of what the numbers do not mean. The modern world is an ocean of data. Waves of information buffet us from every direction. Our students need to learn how to obtain, analyze, synthesize, evaluate, and draw inferences from statistics.

Online Resources for High School Math

The free resources in this section will appeal to unschoolers and independent learners who want to dabble a bit in the different areas of high school math before committing themselves to deeper study.

Other families, who prefer a traditional approach, may also find these links useful. There will be times when a textbook is not enough. Your student may read the text (or watch the video) several times and work through every sample problem, yet he or she may still have trouble understanding a concept. In cases like that, it helps to look up a different explanation that can give your student a new way of looking at the topic.

Whichever category describes your family, keep in mind what Albert Einstein said: “Learning is experience. Everything else is just information.”

In math, experience means wrestling with problems. Even the most useful website can only provide information. If we want our children to truly learn mathematics, they will have to build their own experience by working their way through lots and lots of math problems.

Algebra

Don’t let your students flounder in algebra: the web is overflowing with help. For instance, many students need hands-on exploration to help them wrap their brains around a new math concept. Henri Picciotto offers a selection of interesting activities at A New Algebra.

Stretch your mathematical modeling skills with the Visual Patterns blog. Pick any design you like and practice recognizing, describing, and predicting the pattern.

Do the explanations in your math book go right over your students’ heads? One of my favorite resources for quick explanations of pre-algebra and algebra topics is Purplemath.

Art of Problem Solving offers a terrific resource with their video lessons featuring Richard Rusczyk, who is always fun to watch. And be sure to take advantage of their online learning system, Alcumus, where students can practice their problem-solving skills.

Geometry

The best way to learn geometry is to play with geometric objects such as lines, curves, angles and shapes. The Maths Is Fun site offers a quick overview of geometry topics, including vocabulary and how to draw various shapes. Or check out the math software GeoGebra, with its wealth of user-created instructional materials for all ages.

For more than two millenia, people have struggled with and mastered Euclidean geometry as the foundation for higher math and logic. It’s still a great way for students to sharpen their thinking skills — as long as they approach each theorem as a puzzle to understand rather than as a rule to memorize. Your students can go straight to the source: Interactive Euclid’s Elements. Or they can work through the easier presentation at Math Open Reference.

The great advances of modern math and science began when geometry and algebra learned to dance together on a coordinate grid. Develop a beginner’s intuition about the meaning of graphs with Graphing Stories. Then explore the relationships between equations and shapes with the Desmos Graphing Calculator, and try your hands at some of the Daily Desmos blog challenges.

Statistics

Many homeschoolers have enjoyed the classic How to Lie With Statistics by Darrell Huff: “The crooks already know these tricks; honest men must learn them in self-defense.” The book is older than I am (in the first chapter, $25,000 is considered a salary that a Yale graduate might boast about), but it is still in print, and the information is as relevant as your daily newspaper. Maybe more so, since newspapers routinely commit many of the errors Huff warns against.

For a basic introduction online, check out Robert Niles’s Statistics Every Writer Should Know, a tutorial for math-phobic journalists. Your students may also enjoy browsing the Gallery of Data Visualization, which features several of the world’s best (and worst) statistical graphs. As the authors point out, seeing may or may not be believing, but “above all, data analysis involves visual, as well as statistical, understanding.”

And if your students want to dig deeper into the topic, Stat Trek offers a good Statistics and Probability Tutorial.

Are You Ready for a Textbook?

A mathematics textbook can be a wonderful learning tool for students who know how to use it. A good textbook offers clear explanations of mathematical concepts and an assortment of worked-out sample problems to demonstrate procedures. It can help students become fluent in the language of math, how to represent their thoughts so others can understand them.

Most important, a textbook provides a rich source of practice problems, arranged in order of increasing difficulty, which give students plenty of opportunity to gain the kind of experience that builds true mathematical learning.

Don’t be fooled by your own experience of dry or tedious math classes: textbook mathematics is still math the mathematician’s way, as mental play. But it is no longer the play of a child dabbling in the shallows, nor the play of the couch potato who wants only to observe.

No, this is the play of the athlete, who works hard at training and enjoys seeing his muscles grow firm, who can’t wait to test himself against a new and challenging opponent. As the athlete does not simply memorize rules and procedures, but instead analyzes the game until it truly makes sense and practices every move, no matter how difficult, until it feels natural — so the student must work with each math concept until it becomes a part of himself.

Anything else, as Einstein said, is merely information.

Learning How To Learn Math

I won’t offer specific advice about choosing your high school math curriculum. Each family is different, and even within a family, what works for one child may fail miserably with another. After 25 years of homeschooling, I’m still reduced to guessing which program will interest my youngest daughter and then adjusting my expectations as we work our way through the book.

But whatever math program you end up using, here are a couple of valuable hints:

• Don’t read a lesson straight through, as if it were a novel.
• Do attempt each sample problem yourself, before reading the book’s solution.

For more tips on mastering high school and college mathematics, check out Stan Brown’s detailed instructions about How to Read a Math Book and How to Study Math at the Tompkins Cortland Community College website.

Working Yourself Out of a Job

If your students have a strong foundation of understanding elementary and middle-school math, and if they are mature enough to recognize when they need to ask for help, they might be able to do their high school math independently. You may find yourself serving more as occasional consultant than as daily instructor.

Even if you never studied trigonometry, for instance, a few Socratic questions will often lead your student toward the answer:

• Describe your problem to me.
• What do you know?
• What information do they give you?
• What do you want?
• What are you trying to find?
• What can you do?
• Tell me what you’ve already tried.
• Have you had a problem like this one before?
• Let’s look at the examples in the chapter …

When I first studied calculus, my dad helped me through at least one killer problem in every homework set this way, just by asking questions. He later admitted to me that he truly hadn’t known what I was supposed to be doing, but I never would have guessed that. His questions were enough to spark my memory of something we had studied in class or of an earlier problem that helped me work through the place I was stumped.

You will find more tips on solving high school math problems in my blog post The Case of the Mysterious Story Problem. For assistance with specific questions about high school math, be sure to check out the Forums Where You Can Ask for Help section on my Internet Math Resources page.

Best wishes to you and your children as you continue to pursue the lifelong adventure of learning!

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This post is an excerpt from my book Let’s Play Math: How Families Can Learn Math Together—and Enjoy It, now available at your favorite online book dealer.