Learning, Not Tedium
I have intended to write more about our home schooling on this blog, so I'm playing catch-up.
Education critics (usually outside of the field of education) often throw around terms like "fuzzy math," or "new math," and I wonder if they have ever really examined the curriculum and approaches recommended by the National Council of Teachers of Mathematics (NCTM). When children are taught using manipulatives, the goal is for understanding of the concept, not just memorization of facts and algorithms. The idea of memorization is not excluded, though. At some point, for efficiency and accuracy, students must memorize their basic math facts. The goal, though, is for the memorization to take place AFTER they understand what those facts really mean. Practice also has its place. Once a student understands a concept, it is important that he practice it, especially in the context of real-world problems.
Here's an example from the past: Even and Odd numbers
This is the only time I would expect my child to do this activity. It's not something I would repeat exactly, although we would do similar portions in the context of multiplication and division. Mathematics is about patterns and this activity allowed the child to see the pattern as it developed, both through the manipulatives and through symbolic representation on the hundreds chart.
Starting with the number 1, she had to "make" each number out of cubes. The pattern is easy to spot and at some point this activity also became practice. The number 1 has only one cube; there aren't enough to make a pair. The number 2, however, can be made with a pair (an "even" pair), with no single, or "odd" cubes left out. Then, the number 3 is formed with a pair of cubes and an odd cube remaining, and so on... Each discovery was recorded on the number chart as the even numbers were colored red.
On the chart, patterns emerged as to the even numbers. Then, I took the chart away and randomly called out numbers. The child had to form the number with cubes, determine if it was odd or even and explain why. "Prove it" is our favorite math phrase!
So, compare this activity to giving a worksheet and telling the student:
1. Numbers are odd or even.
2. Even numbers are: 2, 4, 6, 8, and all the numbers that end in these numbers. Oh, and zero, also.
3. If a number is not even, it is odd. Or, if a number is not divisible by 2, it is odd for older students.
4. Do these worksheets. (Circle the even numbers; finish the pattern, etc...)
Now, do I expect my children to carry cubes around with them and figure out if a number is even or odd by forming pairs with cubes? Absolutely not; this activity was about learning the concept. It will give the child a concrete experience with the concept of even and odd. They naturally discover what are the even and odd numbers and they recognize the pattern. They will then be able to apply that pattern to larger numbers in other contexts.
For students that you suspect have this concept mastered, you could use this activity as an assessment, asking them to form the numbers and explain to you which ones are even or odd and why. You would want to ask those questions in either case. We have to be careful about judging mastery of the concept though.
Some students, given this pattern to complete: 2, 4, 6, 8, 10, ____, ____, ____ may be able to complete the pattern correctly, but may have done so because they recognized the "skip counting" pattern, but they have no understanding of the numbers being divisible by two. The idea that even numbers are multiples of two is the concept you want the child to understand here. For this and other patterns, you want a child to be able to:
1. Recognize the pattern.
2. Communicate the rule, or what is happening, in the pattern, in words (orally is fine) and in symbols.
3. Determine the next components of the pattern.
All math instruction is about building--one concept upon another. Concrete activities like this help lay a solid foundation that the student may build upon. Also, imagine learning as a series of hooks in the brain. The brain looks for new ideas to hook upon from previous experience. This gives a hook to build upon, so the information can truly be learned. Simple memorization of even and odd numbers after only symbolic exposure to the numbers could result in a foundation on sand that won't allow more building to continue, causing the whole mathematical structure to collapse.
Education critics (usually outside of the field of education) often throw around terms like "fuzzy math," or "new math," and I wonder if they have ever really examined the curriculum and approaches recommended by the National Council of Teachers of Mathematics (NCTM). When children are taught using manipulatives, the goal is for understanding of the concept, not just memorization of facts and algorithms. The idea of memorization is not excluded, though. At some point, for efficiency and accuracy, students must memorize their basic math facts. The goal, though, is for the memorization to take place AFTER they understand what those facts really mean. Practice also has its place. Once a student understands a concept, it is important that he practice it, especially in the context of real-world problems.
Here's an example from the past: Even and Odd numbers
This is the only time I would expect my child to do this activity. It's not something I would repeat exactly, although we would do similar portions in the context of multiplication and division. Mathematics is about patterns and this activity allowed the child to see the pattern as it developed, both through the manipulatives and through symbolic representation on the hundreds chart.
Starting with the number 1, she had to "make" each number out of cubes. The pattern is easy to spot and at some point this activity also became practice. The number 1 has only one cube; there aren't enough to make a pair. The number 2, however, can be made with a pair (an "even" pair), with no single, or "odd" cubes left out. Then, the number 3 is formed with a pair of cubes and an odd cube remaining, and so on... Each discovery was recorded on the number chart as the even numbers were colored red.
 | |
| An even number can be made with pairs of 2 with no odd cubes left out |
 | ||||
| An odd number will always have one odd cube left out after all the pairs are formed |
So, compare this activity to giving a worksheet and telling the student:
1. Numbers are odd or even.
2. Even numbers are: 2, 4, 6, 8, and all the numbers that end in these numbers. Oh, and zero, also.
3. If a number is not even, it is odd. Or, if a number is not divisible by 2, it is odd for older students.
4. Do these worksheets. (Circle the even numbers; finish the pattern, etc...)
Now, do I expect my children to carry cubes around with them and figure out if a number is even or odd by forming pairs with cubes? Absolutely not; this activity was about learning the concept. It will give the child a concrete experience with the concept of even and odd. They naturally discover what are the even and odd numbers and they recognize the pattern. They will then be able to apply that pattern to larger numbers in other contexts.
For students that you suspect have this concept mastered, you could use this activity as an assessment, asking them to form the numbers and explain to you which ones are even or odd and why. You would want to ask those questions in either case. We have to be careful about judging mastery of the concept though.
Some students, given this pattern to complete: 2, 4, 6, 8, 10, ____, ____, ____ may be able to complete the pattern correctly, but may have done so because they recognized the "skip counting" pattern, but they have no understanding of the numbers being divisible by two. The idea that even numbers are multiples of two is the concept you want the child to understand here. For this and other patterns, you want a child to be able to:
1. Recognize the pattern.
2. Communicate the rule, or what is happening, in the pattern, in words (orally is fine) and in symbols.
3. Determine the next components of the pattern.
All math instruction is about building--one concept upon another. Concrete activities like this help lay a solid foundation that the student may build upon. Also, imagine learning as a series of hooks in the brain. The brain looks for new ideas to hook upon from previous experience. This gives a hook to build upon, so the information can truly be learned. Simple memorization of even and odd numbers after only symbolic exposure to the numbers could result in a foundation on sand that won't allow more building to continue, causing the whole mathematical structure to collapse.
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