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Four principles of deeply effective math teaching

If you were asked what were the most important principles in mathematics teaching, what would you say? I wasn't really asked, but I started thinking, and came up with these basic habits that can keep your math teaching on the right track.

Principle 1: Let It Make Sense

Let us strive to teach for understanding of mathematical concepts and procedures, the "why" something works, and not only the "how".

This understanding, as I'm sure you realize, doesn't always come immediately. It may take even several years to grasp a concept. For example, place value is something children understand partially at first, and then that deepens over a few years.

This is why many math curricula use spiraling: they come back to a concept the next year, the next year, and the next. This can be very good if not done excessively (for 5-6 years is probably excessive).

However, spiraling has pitfalls also: if your child doesn't get a concept, don't blindly "trust" the spiraling and think, "Well, she gets it the next year when the book comes back around to it."

The next year's schoolbook won't necessarily present the concept at the same level - the presentation might be too difficult. If a child doesn't "get it", they might need very basic instruction for the concept again.

The "how" something works is often called procedural understanding: the child knows how to work long division or knows the procedure for fraction addition. It is often possible to learn the "how" mechanically without understanding why something works. Procedures learned this way are often forgotten very easily.

The relationship between the "how" and the "why" - or between procedures and concepts - is complex. One doesn't always come totally before the other, and it also varies from child to child. And, conceptual and procedural understanding actually help each other: conceptual knowledge (understanding the "why") is important for the development of procedural fluency, while fluent procedural knowledge supports the development of further understanding and learning.

Try alternating the instruction: teach how to add fractions, and let the student practice. Then explain why it works. Go back to some practice. Back and forth. Sooner or later it should 'stick' - but it might be next year instead of this one, or after 6 months instead of this month.

As a rule of thumb, don't totally leave a topic until the student both knows "how" and understands the "why".

Tip: you can often test a student's understanding of a topic by asking him to produce an example, preferably with a picture or other illustration: "Tell me an example of multiplying a fraction by a whole number, and draw a picture of it." Whatever gets produced can tell the teacher a lot about what has been understood.

Principle 2: Remember the Goals

What are the goals of your math teaching? Are they...

• to finish the book by the end of school year
• make sure the kids pass the test ...?

Or do you have goals such as:

• My student can add, simplify, and multiply fractions
• My student can divide by 10, 100, and 1000.

These are all just "subgoals". But what is the ultimate goal of learning school mathematics?

Consider these goals:

• Students need to be able to navigate their lives in this ever-so-complex modern world.
This involves dealing with taxes, loans, credit cards, purchases, budgeting, and shopping. Our youngsters need to be able to handle money wisely. All that requires good understanding of parts, proportions, and percents.

• Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, and percents.

• And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.

• I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course high school geometry is a good example of this, but when taught properly, other areas of school math can be as well.

• Then one more goal that I personally feel fairly strongly about: let students see some beauty of mathematics and to learn to like it, or at the very least, make sure they don't feel negatively about mathematics.

The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.

For example, adding, simplifying, and multiplying fractions all connect with the broader goal of understanding part-and-whole relationships. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a necessary basis for solving rational equations and for the operations with rational expressions (in algebra).

Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals — not a goal in itself. Don't ever be a slave to any math book.

A math teacher's tools are quite numerous nowadays.

First of all of course comes a black or white board or paper — something to write on, then we have pencils, compass, protractor, ruler, eraser....
And the book you're using.

Then we have computer software, interactive activities, animated lessons and such.

There are workbooks, fun books, worktexts, books, and online tutorials.

Then we have manipulatives, abacus, measuring cups, scales, algebra tiles, and so on. And then there are games, games, games.

The choices are so numerous it's daunting. What's a teacher to do?

Well, you just have to start somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.

There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.

Basic tools

1. The board and/or paper to write on. Essential. Easy to use.
2. The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check our curriculum pages for some help. There are two things to keep in mind:
1. No matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
2. Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.
3. Manipulatives are physical objects the student manipulates with his hands to get a better grasp of some concept.

I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a "must".

Manipulatives are definitely stressed in these days. They are usually very recommendable, but they're not the final goal of math education, and there is no need to over-emphasize them. The goal is to learn to do the math without them.

• base ten blocks or something to illustrate tens & ones in kindergarten and first grade. I made my daughter "ten-bags" by putting marbles into little plastic bags, and they worked perfectly for teaching place value.
• some kind of fraction manipulatives. You can simply make pie models out of cardboard.

Often, drawing pictures can take place of manipulatives, especially after the first elementary grades.

4. Geometry and measuring tools, such as ruler, compass, protractor, scales, and measuring cups. These are of course essential teaching tools. (Note though that dynamic geometry software can in these days replace compass and ruler constructions done on paper and actually be even better.)

The extras

These are, obviously, too many to even start listing.

Principle 4: Living and Loving Math

You are the teacher. You show the way - also with your attitudes, your way of life.

Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?

And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?

Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your children a way of life and whether math is a natural part of it or not.

Math is not a drudgery, nor something just confined to math lessons.

Some ideas:

• Let it make sense. This alone can usually make quite a difference and students will stay interested.
• Read through some fun math books, such as Theoni Pappas books or puzzle books. Get to know some interesting math topics besides just schoolbook arithmetic. There are lots of story books (math readers) that teach math concepts - see a list here.
• Consider including some math history if you have the time.
• When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.