### 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.

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 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

## Comments

In my finger math tutorial, it takes seven lessons to get to single digit addition. However, in one additional lesson, i show how to speed up single digit addition, with the introduction of a new concept. Mastery of this technique not only speeds addition, but it improves reliability by eliminating a step. This eliminates having to remember to do something later. Having to remember something is, in my opinion, why carries and borrows contribute to failure in addition and subtraction. Indeed, i blame them for fully half of all fear of math. It's not division, or fractions, or algebra. Off by one is still wrong.

Part six (which is the seventh lesson) is up on my blog:

http://predelusional.blogspot.com/2006/06/finger-arithmetic-part-six.html

I guess you'll be able to see what i'm talking about when part seven is posted - next week.