This is part five in a six-part series on the development of number sense. In part one, I outlined the importance of developing a sense for numbers that serves as
a foundation upon which we build our mathematical structure. In the remaining installments I addressed five components of number sense.
· Number magnitude
· Estimation
· Mental computation
· Mathematical properties
· Effects of operations
In this blog, I will talk about the mathematical properties. Typically, our state test only assesses students on the names of these properties. For example, it might ask students to identify the commutative property of addition and offer a+b=b+a as one choice among four on a test. This type of question determines whether the student has memorized the property in it’s abstract form but does not determine whether the student knows how to apply it. Since a proficiency with number requires that students know how to apply formulas to help them mathematically, I will focus on that subject here.
Often I will ask my students to solve a math problem that appears difficult unless they correctly apply the property. For example, I might ask them to mentally solve this problem:
37 + 256 + 19 + 50 =
Students who obediently work left to right are likely to lose track of the partial sums before they finish the problem. However applying the associative property of addition allows us to rearrange the addends this way:
37+19+(44+6)+256=
(37+19+44)+(6+256)=
100+262=
362
I once saw a student solve 18÷1.5 very quickly. He said, “Thirty-six divided by three is twelve.” He couldn’t explain why that worked, but he knew that it would. This gave me the opportunity to show the class that he had used the multiplicative identity (1) to simplify the problem by multiplying the numerator and denominator by 2. Since 2/2 = 1, he had not changed the problem.
I might ask students to solve this problem:
6(254)
This might be challenging if we use the algorithm traditionally taught in which a student begins on the right side and used regrouping as they work. However, the distributive property allows us to begin with the most significant digits on the left side as we work toward the right. This makes the mental addition simpler.
6(254) = 6(200 + 50 + 4) = 1200 + 300 + 24 = 1524
Another time I asked teachers at a math training to mentally solve:
10,002–4,566
With the borrowing model taught in the United States this may prove difficult. However some countries use a compensation model of subtraction to solve this. Though unusual for us, the strategy is easy to implement. In such nations, one would simply sovle:
9,999–4,563=5,436
The idea is that there is a specific difference between 10,002 and 4,566. If we subtract three from both of those numbers the difference between them remains the same and the subtraction requires no regrouping. You might think about two people who have those amounts of money. One has $5,436 more. If they both spend $3, then the one still has $5,436 more.
Knowing how to use mathematical properties is of greater value than knowing their names. However, students are tested on mathematical vocabulary as well as the application. For this reason, I teach students how to apply the properties, and then I teach them the names of those properties.
Textbooks usually offer a new term, define it, and give some examples prior to asking students to apply it. However this is not how we learn terms outside of a classroom. If I want to take up golf, I don’t have to learn the language first. I pick up terms like hook and slice as my amateurish shots careen off the sides of the fairway.
Once students are familiar with applying a property to make sense of their mathematics, I then introduce the term by saying, “You know, there’s a word for that…” This then allows them not only to simplify their calculations with this new property, they can simplify their discussion of it with this new term.
· Number magnitude
· Estimation
· Mental computation
· Mathematical properties
· Effects of operations
In this blog, I will talk about the mathematical properties. Typically, our state test only assesses students on the names of these properties. For example, it might ask students to identify the commutative property of addition and offer a+b=b+a as one choice among four on a test. This type of question determines whether the student has memorized the property in it’s abstract form but does not determine whether the student knows how to apply it. Since a proficiency with number requires that students know how to apply formulas to help them mathematically, I will focus on that subject here.
Often I will ask my students to solve a math problem that appears difficult unless they correctly apply the property. For example, I might ask them to mentally solve this problem:
37 + 256 + 19 + 50 =
Students who obediently work left to right are likely to lose track of the partial sums before they finish the problem. However applying the associative property of addition allows us to rearrange the addends this way:
37+19+(44+6)+256=
(37+19+44)+(6+256)=
100+262=
362
I once saw a student solve 18÷1.5 very quickly. He said, “Thirty-six divided by three is twelve.” He couldn’t explain why that worked, but he knew that it would. This gave me the opportunity to show the class that he had used the multiplicative identity (1) to simplify the problem by multiplying the numerator and denominator by 2. Since 2/2 = 1, he had not changed the problem.
I might ask students to solve this problem:
6(254)
This might be challenging if we use the algorithm traditionally taught in which a student begins on the right side and used regrouping as they work. However, the distributive property allows us to begin with the most significant digits on the left side as we work toward the right. This makes the mental addition simpler.
6(254) = 6(200 + 50 + 4) = 1200 + 300 + 24 = 1524
Another time I asked teachers at a math training to mentally solve:
10,002–4,566
With the borrowing model taught in the United States this may prove difficult. However some countries use a compensation model of subtraction to solve this. Though unusual for us, the strategy is easy to implement. In such nations, one would simply sovle:
9,999–4,563=5,436
The idea is that there is a specific difference between 10,002 and 4,566. If we subtract three from both of those numbers the difference between them remains the same and the subtraction requires no regrouping. You might think about two people who have those amounts of money. One has $5,436 more. If they both spend $3, then the one still has $5,436 more.
Knowing how to use mathematical properties is of greater value than knowing their names. However, students are tested on mathematical vocabulary as well as the application. For this reason, I teach students how to apply the properties, and then I teach them the names of those properties.
Textbooks usually offer a new term, define it, and give some examples prior to asking students to apply it. However this is not how we learn terms outside of a classroom. If I want to take up golf, I don’t have to learn the language first. I pick up terms like hook and slice as my amateurish shots careen off the sides of the fairway.
Once students are familiar with applying a property to make sense of their mathematics, I then introduce the term by saying, “You know, there’s a word for that…” This then allows them not only to simplify their calculations with this new property, they can simplify their discussion of it with this new term.
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