Encouraging Mathematical Thinking
Encouraging Mathematical Thinking
July 31, 2001
Does the thought of helping your child learn math conjure up
visions of flash-card drills and timed tests of <+>the basic
facts<+>? Any four-function calculator can quickly add,
subtract, multiply, or divide numbers. Far more important to your
child's mathematical development is the ability to pose and solve
problems, to make reasonable estimates, and to communicate
solutions.
How do children become good at mathematics? The answer is simple--by using mathematics in their daily lives. There are many things parents can do that will help their children develop a disposition for and an interest in mathematics, according to Dr. Susan Taber, chairperson of the Elementary/Early Childhood Education Department at Rowan University, Glassboro. Taber, educated at Stanford University and the University of Delaware, suggests:
* Encourage the posing and solving of problems. Mathematics is the means by which human beings quantify and describe the world. Many activities have mathematical components. Parents can help their children pose and solve problems based on their daily activities. For example: <+>If there are 12 dinner plates in the cupboard, how many will be left in the cupboard after setting the table for dinner?<+> <+>There are 14 steps from the first floor to the second; how many steps are left to be climbed when you are on the fifth one?<+> <+>If our family is going to share one dozen doughnuts, what is one person's fair share?<+> <+>If there are 10 hot dogs in one package, how many are there in three packages?<+> <+>How much will it cost our family to go to the movies?<+>
* A collection of objects such as a box of discarded buttons, a set of expired grocery coupons, or a jar of loose change provides a wealth of problem-solving opportunities. After the child has sorted and classified the objects, you can ask questions such as: <+>How many more red buttons are there than blue buttons?<+> <+>How dimes and nickels are there altogether?<+> <+>How much money could be saved by using all the 25-cent coupons at the grocery store?<+> <+>Find some coins that add up to 75 cents.<+> Asking the child to explain how he/she arrived at his/her answer encourages him/her to examine and clarify his/her mathematical thinking. You also can encourage your child to write down the problem and its solution using pictures, diagrams, or numerals.
* Older children can be encouraged to figure out how long it will take the family to arrive at a destination if the car is traveling at 50 miles per hour. You can encourage your child to predict the time of arrival and compare it with the actual time of arrival.
* You can also encourage your child to make up problems for you to solve. This will give you insights into the magnitudes of numbers that are meaningful to your child, as well as his/her understanding of mathematical actions such as combining, comparing, taking away, or sharing quantities. If the numbers the child includes in a problem are too large for the child to compute, encourage him/her to use a calculator to solve the problem.
* Going beyond counting. Children, even before receiving any school instruction, can solve many kinds of mathematical problems. Generally, they do so by modeling the quantities in the problem with their fingers or counters (if available), and then adding to or taking away the required amount to solve the problem. While modeling and counting can be an effective way of solving problems like 9 + 4, it becomes less accurate and more cumbersome when larger amounts are involved. Unfortunately, many children continue to count by ones when adding larger numbers such as 29 + 35 because it seems to them to be the most reliable method of computing the sum. You can help your child develop more efficient and reliable methods for computation by helping him/her to develop reliable strategies that are more accurate and more efficient for the student to use.
* The first time- and energy-saving strategy is to recognize that numbers can be added in any order. Students who are stuck in counting often don't realize that 8 + 5 has the same answer as 5 + 8. Once students understand and accept the order property (sometimes called the commutative property), the work of remembering addition combinations is immediately cut by 50 percent. There are many simple games that help students understand this principle. For example, have your child use objects of two different colors to explore all the ways that two numbers can add up to 12. As the child makes each combination, he/she should write it down on a piece of paper. As he/she looks for patterns in his/her list of combinations, he/she will notice that 4 + 8 and 8 + 4 both have the sum of 12, as do other pairs such as 5 + 7 and 7 + 5. The student can place 12 objects in a cup and spill the objects out onto a piece of paper. Some of the objects will land on the paper and some will land off the paper. Try to get him/her to count the objects that have landed <+>on<+> the paper and then say how many landed <+>off<+> the paper.
* The second strategy for remembering addition combinations is to use doubles. The human brain remembers the sums of doubles more easily than other combinations. When the child is confident that 4 + 4 = 8 or 6 + 6 = 12, he/she has a powerful tool for learning more difficult combinations such as 4 + 5, 5 + 6, and 6 + 7. It takes very little time to think of 5 + 6 as one more than 5 + 5 = 10. Interestingly enough, the most difficult addition combinations such as 7 + 6, 7 + 8, 8 + 9 are near doubles.
* The third strategy is to help the child understand that numbers can be broken apart and recombined in easier combinations. For example, 8 + 4 can be thought of as 8 + 2 and 2 more. Similarly, another difficult combination 8 + 5 can be thought of as 8 + 2 and 3 more. Learning the combinations that make 10 provides a powerful means of turning difficult combinations such as 9 + 6, 9 + 7 into easily computed sums.
* Foster the development of number sense. If you wanted to buy an item costing $1.50 but had only 98 cents, you'd probably figure out how much more you needed by thinking of adding two cents to the 98 to get to one dollar and then adding 50 cents more to arrive at $1.50. This is much easier than setting up a subtraction problem which requires borrowing (or more correctly, regrouping) twice to ?subtract 8 from 0? and then ?subtract 9 from 14.? Encourage your child to develop ways of adding and subtracting that make sense to him/her. Children who understand the meaning of the digits in two- and three-digit numerals prefer to add the <+>tens<+> or the <+>hundreds<+> first, before adding the ones. For example, when adding 28 + 24, it is just as accurate to add 20 + 20 and then add on the sum of 8 and 4 as it is to add 4 + 8 first. Adding 20 to 28 to obtain 48 and then adding on the <+>4<+> will also give the correct answer of 52. Instead of insisting that your child use the standard <+>regrouping<+> algorithms for addition and subtraction, ask your child to solve problems in ways that make sense to him/her and then to explain his/her methods. After all, the goal is to find a correct answer using a method that makes sense and is reliable, not to memorize a particular procedure.
* Another enjoyable and productive activity is to ask the members of the family to come up with as many combinations as they can that equal a given number. For example, in addition to identifying 5 + 20 and 15 + 10 as combinations that equal 25, your child may eagerly come up with combinations such as 19 + 6, 30 - 5, 100/4, and many others. Any number can serve as the base number. As your child searches for novel ways to represent that number, he/she will not only get lots of computation practice but also will make many discoveries about the structure of our number system and of the relationships among numbers in the system.
* Give your child authentic experiences with the tools of our civilization. Over the course of human history, we have developed many methods of measuring and counting quantities. Although we write monetary amounts in base-ten notation, coins and bills do not always conform to the base-ten system: e.g., nickels, quarters, half-dollars, two-dollar bills, twenties, and so on. Although we write numbers using the base-ten place value system, we use a combination of base 12 and base 60 systems for representing time. We measure length using a combination base 12 and base 3 system. We measure volume using measures that increase by factors of 2 (cups to pints and pints to quarts), of 4 (cups to quarts and quarts to gallons) and of 3 (teaspoons to tablespoons).
* Giving your child the opportunity to measure objects while cooking or making crafts not only will help your child make sense of these various systems but also will help the child learn fraction relationships and combinations. Ask your child to help you figure out how much is needed of each ingredient to double or triple a recipe. Most food packages now give both the metric (grams, liters, kilograms) and the customary (ounces, quarts, pounds) measures of the contents. Challenge your child to figure out the size of a serving from the information on the package or, alternatively, the number of servings of a given size contained in the package using both metric and customary measures. If you have a digital clock and an analog clock (with hands) in the same room be sure that they are set to exactly the same time. Let your child figure out how she could be 120 or you could be 480. As your child solves problems like this, he/she will not only become conversant with our various systems of measurement but also will gain important understandings related to proportions, ratios, and fractions. Children who have authentic experiences using money, clocks, measuring cups, and rules generally learn these topics more easily than children who have not had such experiences.
How do children become good at mathematics? The answer is simple--by using mathematics in their daily lives. There are many things parents can do that will help their children develop a disposition for and an interest in mathematics, according to Dr. Susan Taber, chairperson of the Elementary/Early Childhood Education Department at Rowan University, Glassboro. Taber, educated at Stanford University and the University of Delaware, suggests:
* Encourage the posing and solving of problems. Mathematics is the means by which human beings quantify and describe the world. Many activities have mathematical components. Parents can help their children pose and solve problems based on their daily activities. For example: <+>If there are 12 dinner plates in the cupboard, how many will be left in the cupboard after setting the table for dinner?<+> <+>There are 14 steps from the first floor to the second; how many steps are left to be climbed when you are on the fifth one?<+> <+>If our family is going to share one dozen doughnuts, what is one person's fair share?<+> <+>If there are 10 hot dogs in one package, how many are there in three packages?<+> <+>How much will it cost our family to go to the movies?<+>
* A collection of objects such as a box of discarded buttons, a set of expired grocery coupons, or a jar of loose change provides a wealth of problem-solving opportunities. After the child has sorted and classified the objects, you can ask questions such as: <+>How many more red buttons are there than blue buttons?<+> <+>How dimes and nickels are there altogether?<+> <+>How much money could be saved by using all the 25-cent coupons at the grocery store?<+> <+>Find some coins that add up to 75 cents.<+> Asking the child to explain how he/she arrived at his/her answer encourages him/her to examine and clarify his/her mathematical thinking. You also can encourage your child to write down the problem and its solution using pictures, diagrams, or numerals.
* Older children can be encouraged to figure out how long it will take the family to arrive at a destination if the car is traveling at 50 miles per hour. You can encourage your child to predict the time of arrival and compare it with the actual time of arrival.
* You can also encourage your child to make up problems for you to solve. This will give you insights into the magnitudes of numbers that are meaningful to your child, as well as his/her understanding of mathematical actions such as combining, comparing, taking away, or sharing quantities. If the numbers the child includes in a problem are too large for the child to compute, encourage him/her to use a calculator to solve the problem.
* Going beyond counting. Children, even before receiving any school instruction, can solve many kinds of mathematical problems. Generally, they do so by modeling the quantities in the problem with their fingers or counters (if available), and then adding to or taking away the required amount to solve the problem. While modeling and counting can be an effective way of solving problems like 9 + 4, it becomes less accurate and more cumbersome when larger amounts are involved. Unfortunately, many children continue to count by ones when adding larger numbers such as 29 + 35 because it seems to them to be the most reliable method of computing the sum. You can help your child develop more efficient and reliable methods for computation by helping him/her to develop reliable strategies that are more accurate and more efficient for the student to use.
* The first time- and energy-saving strategy is to recognize that numbers can be added in any order. Students who are stuck in counting often don't realize that 8 + 5 has the same answer as 5 + 8. Once students understand and accept the order property (sometimes called the commutative property), the work of remembering addition combinations is immediately cut by 50 percent. There are many simple games that help students understand this principle. For example, have your child use objects of two different colors to explore all the ways that two numbers can add up to 12. As the child makes each combination, he/she should write it down on a piece of paper. As he/she looks for patterns in his/her list of combinations, he/she will notice that 4 + 8 and 8 + 4 both have the sum of 12, as do other pairs such as 5 + 7 and 7 + 5. The student can place 12 objects in a cup and spill the objects out onto a piece of paper. Some of the objects will land on the paper and some will land off the paper. Try to get him/her to count the objects that have landed <+>on<+> the paper and then say how many landed <+>off<+> the paper.
* The second strategy for remembering addition combinations is to use doubles. The human brain remembers the sums of doubles more easily than other combinations. When the child is confident that 4 + 4 = 8 or 6 + 6 = 12, he/she has a powerful tool for learning more difficult combinations such as 4 + 5, 5 + 6, and 6 + 7. It takes very little time to think of 5 + 6 as one more than 5 + 5 = 10. Interestingly enough, the most difficult addition combinations such as 7 + 6, 7 + 8, 8 + 9 are near doubles.
* The third strategy is to help the child understand that numbers can be broken apart and recombined in easier combinations. For example, 8 + 4 can be thought of as 8 + 2 and 2 more. Similarly, another difficult combination 8 + 5 can be thought of as 8 + 2 and 3 more. Learning the combinations that make 10 provides a powerful means of turning difficult combinations such as 9 + 6, 9 + 7 into easily computed sums.
* Foster the development of number sense. If you wanted to buy an item costing $1.50 but had only 98 cents, you'd probably figure out how much more you needed by thinking of adding two cents to the 98 to get to one dollar and then adding 50 cents more to arrive at $1.50. This is much easier than setting up a subtraction problem which requires borrowing (or more correctly, regrouping) twice to ?subtract 8 from 0? and then ?subtract 9 from 14.? Encourage your child to develop ways of adding and subtracting that make sense to him/her. Children who understand the meaning of the digits in two- and three-digit numerals prefer to add the <+>tens<+> or the <+>hundreds<+> first, before adding the ones. For example, when adding 28 + 24, it is just as accurate to add 20 + 20 and then add on the sum of 8 and 4 as it is to add 4 + 8 first. Adding 20 to 28 to obtain 48 and then adding on the <+>4<+> will also give the correct answer of 52. Instead of insisting that your child use the standard <+>regrouping<+> algorithms for addition and subtraction, ask your child to solve problems in ways that make sense to him/her and then to explain his/her methods. After all, the goal is to find a correct answer using a method that makes sense and is reliable, not to memorize a particular procedure.
* Another enjoyable and productive activity is to ask the members of the family to come up with as many combinations as they can that equal a given number. For example, in addition to identifying 5 + 20 and 15 + 10 as combinations that equal 25, your child may eagerly come up with combinations such as 19 + 6, 30 - 5, 100/4, and many others. Any number can serve as the base number. As your child searches for novel ways to represent that number, he/she will not only get lots of computation practice but also will make many discoveries about the structure of our number system and of the relationships among numbers in the system.
* Give your child authentic experiences with the tools of our civilization. Over the course of human history, we have developed many methods of measuring and counting quantities. Although we write monetary amounts in base-ten notation, coins and bills do not always conform to the base-ten system: e.g., nickels, quarters, half-dollars, two-dollar bills, twenties, and so on. Although we write numbers using the base-ten place value system, we use a combination of base 12 and base 60 systems for representing time. We measure length using a combination base 12 and base 3 system. We measure volume using measures that increase by factors of 2 (cups to pints and pints to quarts), of 4 (cups to quarts and quarts to gallons) and of 3 (teaspoons to tablespoons).
* Giving your child the opportunity to measure objects while cooking or making crafts not only will help your child make sense of these various systems but also will help the child learn fraction relationships and combinations. Ask your child to help you figure out how much is needed of each ingredient to double or triple a recipe. Most food packages now give both the metric (grams, liters, kilograms) and the customary (ounces, quarts, pounds) measures of the contents. Challenge your child to figure out the size of a serving from the information on the package or, alternatively, the number of servings of a given size contained in the package using both metric and customary measures. If you have a digital clock and an analog clock (with hands) in the same room be sure that they are set to exactly the same time. Let your child figure out how she could be 120 or you could be 480. As your child solves problems like this, he/she will not only become conversant with our various systems of measurement but also will gain important understandings related to proportions, ratios, and fractions. Children who have authentic experiences using money, clocks, measuring cups, and rules generally learn these topics more easily than children who have not had such experiences.