by Josh Rappaport
When I work with children on elementary math operations, I often like to bring out what I call the “math menu.”
Instead of showing children only one way to work a math problem, I offer them a "menu" of options, and I tell children that they get to choose the one that works best for them.
Children perk up when they hear that they’re getting a menu of options. Since they get a chance to choose the approach that suits them best, they become more interested. And interest boosts involvement.
Presenting a range of approaches helps children in other ways, too. First, children learn that each option has advantages and disadvantages. So they realize that it might make sense to use Option #1 in one situation, Option #2 in another. Encouraging children to evaluate the relative merits of different options engages them in higher-level thinking skills. And that by itself helps them grow intellectually.
Secondly, encouraging flexibility of thought helps children develop critical thinking skills that serve them in other subjects, too. It could even be said that this helps them in life in general, since life success so often depends on finding the best response out of a range of possible responses.
To give you a sense of what such a math menu looks like, here are three options for teaching subtraction with two-digit numbers.
OPTION #1 — First, there is the "standard approach," in which students re-group, trading one group of 10 for 10 ones. I show students how to move those 10 ones to the 1s place, and then subtract. This is the way I learned to subtract with re-grouping (called ‘borrowing,’ back then in the 'old days').
For example, in the problem 42 – 27, students first demote the 4 of 42 to a 3, then transfer 10 to the 2 of 42, making that 2 into a 12. Students then subtract, first getting: 12 – 7 = 5. Then, after that they subtract in the 10s place, getting: 3 – 2 = 1. Noticing the 1 next to the 5, they see their answer: 15. When teaching this approach the first time, it’s always best to use manipulatives. I have used coins (pennies and dimes), individual and rubber-banded bundles of 10 popsicle sticks, and place-value rods to teach students how to re-group. There are other options available as well for manipulatives.
OPTION #2 — Subtract by "adding up." In the same problem, 42 – 27, I teach students to think of the problem as representing a journey along the number line, with the numbers that are multiples of 10 serving as "towns" along the road where they can stop and get a refreshment. In the problem of 42 – 27, students start out at the lower number, 27, and first travel 3 miles, from milepost 27 to Town 30, where they get a nice drink. Then they travel 10 miles, from Town 30 to Town 40, where they eat lunch. Then they travel 2 more miles, from Town 40 to milepost 42, at which point they reach their destination. Students add up the three distances traveled: 3 + 10 + 2, and they get their answer of 15. This approach lends itself much more to "mental math" — doing the problem in one's mind, without paper and pencil. Since this technique is the one that cashiers used in the past (and some still use), you can let your children learn this technique by having them pretend to be a storekeeper who needs to make change.
OPTION #3 — For my third menu option, I have children subtract from left to right instead of from right to left, the typical approach. Using the left-to-right approach has two big advantages. First, it emphasizes the place values of numbers in the sense that it forces students to think about the place value of digits. For example, in the example below, when children look at the 4 in 42, they need to view it as meaning 40, not just 4. Secondly, this approach offers children a fun introduction to negative numbers. More young children “get” the concept of negative numbers than you might suppose.
Here’s how you teach the left-to-right technique for the same problem: 42 – 27. First have your child subtract in the 10s place: 40 – 20 = 20. Then ask your child to subtract in the 1s place: 2 – 7 = – 5. Putting the two partial answers together, your child should get 20 and – 5. Your child may amaze you and say they realize the answer is 15. Just in case they don’t, though, explain that 20 combined with – 5 is basically the same problem as 20 – 5, which of course equals 15. Give your child more practice problems with this approach, and watch him/her learn how to subtract quickly, correctly and with an understanding of negative numbers. Quite amazing! As a point of interest, this is one of the main approaches used to teach subtraction in British schools, according to several emails I have received.
Some people might think that elementary students are too young to work with negative numbers. But generally I don't need to explain much about negative numbers. I just tell children that when we take more than what we have, the answer is "negative.” Try it yourself and see if your child understands more about numbers than you would have guessed!
In any case, these are the three approaches that I usually present for two-digit subtraction. Of course, there are yet other ways to subtract two-digit numbers from one another (using an abacus is one quick example of another approach). All approaches that work in all situations are mathematically valid.
The thought I'd like to leave you with today is that presenting a menu of options — a range of strategies — can help children find their own preferences for doing math. And when they find their preferences, children enjoy arithmetic more. Give it a try if you have not yet done so. You might want to start with two options and build your way up. As you continue with this approach, you will find new ways to teach the same math skill to your children, and your repertoire of approaches will grow, as will your children’s enjoyment of math.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
When I work with children on elementary math operations, I often like to bring out what I call the “math menu.”
Instead of showing children only one way to work a math problem, I offer them a "menu" of options, and I tell children that they get to choose the one that works best for them.
Children perk up when they hear that they’re getting a menu of options. Since they get a chance to choose the approach that suits them best, they become more interested. And interest boosts involvement.
Presenting a range of approaches helps children in other ways, too. First, children learn that each option has advantages and disadvantages. So they realize that it might make sense to use Option #1 in one situation, Option #2 in another. Encouraging children to evaluate the relative merits of different options engages them in higher-level thinking skills. And that by itself helps them grow intellectually.
Secondly, encouraging flexibility of thought helps children develop critical thinking skills that serve them in other subjects, too. It could even be said that this helps them in life in general, since life success so often depends on finding the best response out of a range of possible responses.
To give you a sense of what such a math menu looks like, here are three options for teaching subtraction with two-digit numbers.
OPTION #1 — First, there is the "standard approach," in which students re-group, trading one group of 10 for 10 ones. I show students how to move those 10 ones to the 1s place, and then subtract. This is the way I learned to subtract with re-grouping (called ‘borrowing,’ back then in the 'old days').
For example, in the problem 42 – 27, students first demote the 4 of 42 to a 3, then transfer 10 to the 2 of 42, making that 2 into a 12. Students then subtract, first getting: 12 – 7 = 5. Then, after that they subtract in the 10s place, getting: 3 – 2 = 1. Noticing the 1 next to the 5, they see their answer: 15. When teaching this approach the first time, it’s always best to use manipulatives. I have used coins (pennies and dimes), individual and rubber-banded bundles of 10 popsicle sticks, and place-value rods to teach students how to re-group. There are other options available as well for manipulatives.
OPTION #2 — Subtract by "adding up." In the same problem, 42 – 27, I teach students to think of the problem as representing a journey along the number line, with the numbers that are multiples of 10 serving as "towns" along the road where they can stop and get a refreshment. In the problem of 42 – 27, students start out at the lower number, 27, and first travel 3 miles, from milepost 27 to Town 30, where they get a nice drink. Then they travel 10 miles, from Town 30 to Town 40, where they eat lunch. Then they travel 2 more miles, from Town 40 to milepost 42, at which point they reach their destination. Students add up the three distances traveled: 3 + 10 + 2, and they get their answer of 15. This approach lends itself much more to "mental math" — doing the problem in one's mind, without paper and pencil. Since this technique is the one that cashiers used in the past (and some still use), you can let your children learn this technique by having them pretend to be a storekeeper who needs to make change.
OPTION #3 — For my third menu option, I have children subtract from left to right instead of from right to left, the typical approach. Using the left-to-right approach has two big advantages. First, it emphasizes the place values of numbers in the sense that it forces students to think about the place value of digits. For example, in the example below, when children look at the 4 in 42, they need to view it as meaning 40, not just 4. Secondly, this approach offers children a fun introduction to negative numbers. More young children “get” the concept of negative numbers than you might suppose.
Here’s how you teach the left-to-right technique for the same problem: 42 – 27. First have your child subtract in the 10s place: 40 – 20 = 20. Then ask your child to subtract in the 1s place: 2 – 7 = – 5. Putting the two partial answers together, your child should get 20 and – 5. Your child may amaze you and say they realize the answer is 15. Just in case they don’t, though, explain that 20 combined with – 5 is basically the same problem as 20 – 5, which of course equals 15. Give your child more practice problems with this approach, and watch him/her learn how to subtract quickly, correctly and with an understanding of negative numbers. Quite amazing! As a point of interest, this is one of the main approaches used to teach subtraction in British schools, according to several emails I have received.
Some people might think that elementary students are too young to work with negative numbers. But generally I don't need to explain much about negative numbers. I just tell children that when we take more than what we have, the answer is "negative.” Try it yourself and see if your child understands more about numbers than you would have guessed!
In any case, these are the three approaches that I usually present for two-digit subtraction. Of course, there are yet other ways to subtract two-digit numbers from one another (using an abacus is one quick example of another approach). All approaches that work in all situations are mathematically valid.
The thought I'd like to leave you with today is that presenting a menu of options — a range of strategies — can help children find their own preferences for doing math. And when they find their preferences, children enjoy arithmetic more. Give it a try if you have not yet done so. You might want to start with two options and build your way up. As you continue with this approach, you will find new ways to teach the same math skill to your children, and your repertoire of approaches will grow, as will your children’s enjoyment of math.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
