Points on math education

I've been lazy lately when it comes to blogging and I'm sorry for that. I've been sort of taking time off from computer work and painting some windows since it just needs done at our house. I view it sort of as "therapy", since it's so different from the computer work and I just take my time and I don't have to think that hard. I like painting.

Anyways, I foud something really nice at Mathmom's. She's written, I feel, an excellent piece about problems in math education in elementary grades. Her point of view is of classroom instruction, but it's still really relevant even for homeschoolers. Some excerpts:

On calculators:

But to be honest, as much as I hate calculator use in school, in this age of calculators and computers, efficiency at hand computation is not, IMO, the most critical math skill for kids to learn. I am NOT saying that it should be ignored, or that kids should be allowed to skip it, and just use calculators in class (see rant linked above). But it is not, IMO, the be all and end all of math education, nor is it a prerequisite, IMO, for studying anything else.

What I consider even more important is a strong sense of number. I want kids who know immediately when the answer they got (either by hand computation or with a calculator) is way off. I want kids who have an instinctive understanding of the distributive law before it is ever formally taught or named (12 sevens is obviously the same as 10 sevens and 2 more sevens). I want kids who know when the amount of change handed to them makes no sense. I would rather have a kid who can multiply 64 x 25 mentally (by halving 64 twice and doubling 25 twice, to see that it's equal to 16 x 100 = 1600) than a kid who can sit down and carry out the long multiplication with pencil and paper, by rote.

I feel that we need to consider several things when it comes to calculators. It's best when the kids can do mental calculations and do paper-and-pencil methods, including understanding why they work. Calculators should be used judiciously, but used. Like she mentions, number sense is of paramount importance so that kids can estimate their answers and tell if the calculator "got it wrong" (e.g. they punched wrong buttons).

On spiraling curricula:

Steve is right that a spiral curriculum can lead to a lax attitude of "it's ok if they don't master this now, because they'll see it again later" that goes on ad infinitum, and the kid never masters anything. This is clearly no good. But the solution isn't necessarily to take away the spiraling for those who need it, IMO. The solution is to have limits - for example, it's ok if they don't completely "get" long multiplication when it's previewed in 3rd grade, or even when it's introduced more formally in 4th, but they have to get it when it's reviewed in 5th, or they shouldn't move on.

This sounds like a really sane approach.

She also talks about including non-routine problems for ALL students to solve. I definitely recommend this practice and have written about it before! MathMom gives several good reasons for this:

1. First, it provides a fabulous way of helping students to appreciate the uses of the procedures and skills they have learned or are learning.

2. Second, this is the kind of thing that "real mathematicians" do! ... A "mathematician" does not sit down and solve 25 ratio and percent word problems, knowing exactly which skills are required to perform the computations. Instead, she investigates "puzzles", looks for interesting patterns makes new discoveries, generalizes results.

3. Third, it develops self esteem and confidence.

4. Fourth, it builds transferrable problem solving skills.

Read it all at Ramblings of a math mom: Math wars.

Points on math education

I've been lazy lately when it comes to blogging and I'm sorry for that. I've been sort of taking time off from computer work and painting some windows since it just needs done at our house. I view it sort of as "therapy", since it's so different from the computer work and I just take my time and I don't have to think that hard. I like painting.

Anyways, I foud something really nice at Mathmom's. She's written, I feel, an excellent piece about problems in math education in elementary grades. Her point of view is of classroom instruction, but it's still really relevant even for homeschoolers. Some excerpts:

On calculators:

But to be honest, as much as I hate calculator use in school, in this age of calculators and computers, efficiency at hand computation is not, IMO, the most critical math skill for kids to learn. I am NOT saying that it should be ignored, or that kids should be allowed to skip it, and just use calculators in class (see rant linked above). But it is not, IMO, the be all and end all of math education, nor is it a prerequisite, IMO, for studying anything else.

What I consider even more important is a strong sense of number. I want kids who know immediately when the answer they got (either by hand computation or with a calculator) is way off. I want kids who have an instinctive understanding of the distributive law before it is ever formally taught or named (12 sevens is obviously the same as 10 sevens and 2 more sevens). I want kids who know when the amount of change handed to them makes no sense. I would rather have a kid who can multiply 64 x 25 mentally (by halving 64 twice and doubling 25 twice, to see that it's equal to 16 x 100 = 1600) than a kid who can sit down and carry out the long multiplication with pencil and paper, by rote.

I feel that we need to consider several things when it comes to calculators. It's best when the kids can do mental calculations and do paper-and-pencil methods, including understanding why they work. Calculators should be used judiciously, but used. Like she mentions, number sense is of paramount importance so that kids can estimate their answers and tell if the calculator "got it wrong" (e.g. they punched wrong buttons).

On spiraling curricula:

Steve is right that a spiral curriculum can lead to a lax attitude of "it's ok if they don't master this now, because they'll see it again later" that goes on ad infinitum, and the kid never masters anything. This is clearly no good. But the solution isn't necessarily to take away the spiraling for those who need it, IMO. The solution is to have limits - for example, it's ok if they don't completely "get" long multiplication when it's previewed in 3rd grade, or even when it's introduced more formally in 4th, but they have to get it when it's reviewed in 5th, or they shouldn't move on.

This sounds like a really sane approach.

She also talks about including non-routine problems for ALL students to solve. I definitely recommend this practice and have written about it before! MathMom gives several good reasons for this:

1. First, it provides a fabulous way of helping students to appreciate the uses of the procedures and skills they have learned or are learning.

2. Second, this is the kind of thing that "real mathematicians" do! ... A "mathematician" does not sit down and solve 25 ratio and percent word problems, knowing exactly which skills are required to perform the computations. Instead, she investigates "puzzles", looks for interesting patterns makes new discoveries, generalizes results.

3. Third, it develops self esteem and confidence.

4. Fourth, it builds transferrable problem solving skills.

Read it all at Ramblings of a math mom: Math wars.

Spiraling or mastery in a mathematics curriculum

Some math curricula are labeled as 'spiraling', and some are said to employ the 'mastery principle'. What does that mean, and which is better?

Spiraling mathematics curriculum introduces many topics in one grade level. It does not aim to teach those topics completely in one go, but revisits those topics the next year, the next year, and so on.

Mastery approach simply aims to teach to mastery any single topic, before going on to the next.

There is a lot of talk against spiraling math curricula, especially in regards to some reformist mathematics curricula.

So is the mastery principle then better?

Well, I think that dividing this matter into two opposite positions is a mistake. You CAN have good mathematics education employing parts of both principles.

For example, a student can learn to add 2-3 digit numbers on grade 2. She can revisit the topic on 3rd grade to learn to add 4-6 digit numbers. She can revisit the topic on 4th grade to learn to add even larger numbers.

At each grade, mastery is required - but in reality it is only partial mastery since there is more to learn about adding numbers.

Or, take fractions. You can visit the topics of fractions on 1st, 2nd, and 3rd grades just passing, noting what is a fraction, maybe having a few addition problems. Then 4th grade some more. Then on 5th grade, you study them a lot, require mastery of certain topics. On 6th, require mastery of all fraction topics. And you're done.

In essence, there is nothing wrong with spiraling. Certainly it is GOOD to present some fraction topics on lower grades, and some later on, revisit the topics next year; or study some long division on 4th and more on 5th. But we need to require mastery of these arithmetic subtopics each year. That way we can at some point easily move into ALGEBRA and leave arithmetic behind.

Yet when it comes to the teaching of math, the divide between the two approaches may be somewhat artificial, as a spiral curriculum could be taught with a judicious eye toward mastery at every level, and any good mastery curriculum will build on concepts in a spiral fashion.
Teens and Tweens Blog, a post on Spiral Curricula (taken offline)

The approach to beware is when a mathematics topic is studied so briefly that kids don't get to master it at all. This can happen with the 'inch-deep-mile-wide' curricula that is so shock-full of topics that all a teacher can do is race thru the book...

See also:
Is your math curriculum coherent and logical?

Scope and sequence chart suggestion

Spiraling or mastery in a mathematics curriculum

Some math curricula are labeled as 'spiraling', and some are said to employ the 'mastery principle'. What does that mean, and which is better?

Spiraling mathematics curriculum introduces many topics in one grade level. It does not aim to teach those topics completely in one go, but revisits those topics the next year, the next year, and so on.

Mastery approach simply aims to teach to mastery any single topic, before going on to the next.

There is a lot of talk against spiraling math curricula, especially in regards to some reformist mathematics curricula.

So is the mastery principle then better?

Well, I think that dividing this matter into two opposite positions is a mistake. You CAN have good mathematics education employing parts of both principles.

For example, a student can learn to add 2-3 digit numbers on grade 2. She can revisit the topic on 3rd grade to learn to add 4-6 digit numbers. She can revisit the topic on 4th grade to learn to add even larger numbers.

At each grade, mastery is required - but in reality it is only partial mastery since there is more to learn about adding numbers.

Or, take fractions. You can visit the topics of fractions on 1st, 2nd, and 3rd grades just passing, noting what is a fraction, maybe having a few addition problems. Then 4th grade some more. Then on 5th grade, you study them a lot, require mastery of certain topics. On 6th, require mastery of all fraction topics. And you're done.

In essence, there is nothing wrong with spiraling. Certainly it is GOOD to present some fraction topics on lower grades, and some later on, revisit the topics next year; or study some long division on 4th and more on 5th. But we need to require mastery of these arithmetic subtopics each year. That way we can at some point easily move into ALGEBRA and leave arithmetic behind.

Yet when it comes to the teaching of math, the divide between the two approaches may be somewhat artificial, as a spiral curriculum could be taught with a judicious eye toward mastery at every level, and any good mastery curriculum will build on concepts in a spiral fashion.
Teens and Tweens Blog, a post on Spiral Curricula (taken offline)

The approach to beware is when a mathematics topic is studied so briefly that kids don't get to master it at all. This can happen with the 'inch-deep-mile-wide' curricula that is so shock-full of topics that all a teacher can do is race thru the book...

See also:
Is your math curriculum coherent and logical?

Scope and sequence chart suggestion