by Maria Miller
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 or principles that can keep your math teaching on the right track.
Habit 1: Let It Make Sense
Habit 2: Remember the Goals
Habit 3: Know Your Tools
Habit 4: Living and Loving Math
Habit 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 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). Many of these principles are used by professionals such as TakeLessons math tutors.
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.
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 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.
Habit 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, 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, 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 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 learn to like it, or at the very least, make sure they do not 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 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 to any math book.
Habit 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.
The choices are so numerous it's daunting. What's a teacher to do?
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.
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
The board and/or paper to write on. Essential. Easy to use.
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:
i) Now 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.
ii) 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.
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
A 100-bead basic 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 after the first few grades and on.
Check out also some virtual manipulatives.
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.
Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the 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 math games and activities online.
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 or principles that can keep your math teaching on the right track.
Habit 1: Let It Make Sense
Habit 2: Remember the Goals
Habit 3: Know Your Tools
Habit 4: Living and Loving Math
Habit 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 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). Many of these principles are used by professionals such as TakeLessons math tutors.
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.
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 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.
Habit 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, 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, 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 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 learn to like it, or at the very least, make sure they do not 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 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 to any math book.
Habit 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.
The choices are so numerous it's daunting. What's a teacher to do?
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.
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
The board and/or paper to write on. Essential. Easy to use.
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:
i) Now 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.
ii) 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.
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
A 100-bead basic 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 after the first few grades and on.
Check out also some virtual manipulatives.
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.
Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the 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 math games and activities online.
Maria Miller has a master's degree in mathematics with the teacher educational studies, and minors in physics and statistics. She loves teaching math. She started writing math books and materials in 2002, after she noticed the poor quality of the materials the homeschooled children that she was tutoring were using. Since then she has been writing books that emphasize conceptual understanding and the "why" of mathematics, while not forgetting the practice needed for mastery.
maria_miller@mathmammoth.com / maria_miller@homeschoolmath.net
