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Four basic principles of deeply effective math teaching


Principle 1: Let It Make Sense
Principle 2: Remember the Goals
Principle 3: Know Your Tools
Principle 4: Living and Loving Math

Principle 1: Let It Make Sense


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 kids 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, and the next. And this can be very good if not done excessively (like for 5-6 years is probably excessive).

However, spiraling also has its own pitfalls: 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 a 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 the procedure of fraction addition or fraction division, for example. It is often possible to learn the "how" mechanically without understanding why it works. Procedures learned this way are often forgotten very easily.


You can try alternating the instruction: teach how to add fractions, and let the student practice. 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 in 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 fractions by whole numbers, and draw a picture." Whatever gets produced can tell the teacher a lot about what has been understood.

Principle 2: Remember the Goals


  • 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


Consider these goals:

    This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely. All that requires good understanding of parts, proportions, and percents.



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


For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).

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 of any math book.

Principle 3: Know Your Tools

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, pencil, compass, protractor, ruler, eraser. And the book you're using. Then we also have computer software, animations and activities online, animated lessons and such. There are workbooks, fun books, worktexts, online texts. Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on. And then there are games, games, games.


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


Basic tools


  2. The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Two things to keep in mind:
  3. Manipulatives. 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' thing.

    Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to go hog wild over them.

    Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.

    Some very helpful manipulatives are

    • abacus
    • something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags.
    • some sort of fraction manipulatives. You can just make pie models out of cardboard, even.

    Often, drawing pictures can take place of manipulatives, especially in the middle grades and on.

    Check out also some virtual manipulatives.

  4. Geometry and measuring tools. These are pretty essential, I'd say. For geometry however, dynamic software can these days replace compass and ruler and easily be far better.

The extras

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

  • Games are nice for about any math topic. I especially recommend games for drilling basic facts. Here's one card game that I played with my daugther when she was learning sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.

  • I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.

  • If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing interactive math games and tutorials online.

Principle 4: Living and Loving Math




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 kids a way of life, and if math is a natural part of it or not.


Some ideas:

  • Let it make sense. This alone can usually make math quite a difference and kids will stay interested.
  • Read through some fun math books, such as Theoni Pappas books or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. There are even story books to teach math concepts.
  • Try including a bit about math history. This might work best in a homeschooling environment where there is no rush to get through the thick book before the year is over.

By Maria Miller

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