by Josh Rappaport
As I said in my last article, math often evokes strong feelings. So why not keep the juices flowing this month, too, by exploring a math issue that evokes passionate debate: the appropriate role of calculators in the homeschooling math curriculum.
Some people feel strongly that calculators have NO PLACE in math studies. On the opposite side, others believe students should have UNRESTRICTED ACCESS to calculators. So who’s right? To whom should you listen?
To me, of course, since I’m writing this article. (Haha!) But given the fact that you have no choice here and now, I’ll try to write this column in as open-minded a way as possible.
Whether you end up agreeing or disagreeing with my position, just thinking about this issue is a good thing, since it gives you a chance to develop or refine your ideas on the topic. For one thing, calculator usage is one of those issues you can’t avoid as a homeschooling parent, unless you live in a yurt and your kids have never even seen a calculator! Also, this is a decision with ramifications; it could very well affect how your children feel about math.
You may be happy to know that I am not an absolutist. That’s in part because, as a tutor for the past 20 or so years, I’ve seen so many different situations regarding kids and math and calculators that I’ve learned that there is no “one-size-fits-all” answer. But I have developed guidelines about calculator use, and that’s what I’ll lay out in this piece.
Before I present my own guidelines, though, let’s take a closer look at some of the arguments by people with strong views, for that gives us important points to consider.
First, some arguments by people who say calculators have ABSOLUTELY NO PLACE in the homeschooling math curriculum.
— Students permitted to use calculators will never memorize critical math facts, like addition or subtraction facts (12 – 5 = 7) or the times table facts (8 x 6 = 48). Or, if they once knew such facts, they will surely forget them, as soon as they become calculator-reliant.
— Calculators give students the message that they no longer have to rely on paper-pencil techniques, or on mental math abilities — to perform mathematical operations. By that I’m referring to operations like 24 x 17, 346 + 986, etc. Students can always “just pull out the calculator“ and start pushing buttons. And as students become more reliant on the calculator, they become weaker at fundamental math skills, such as the core skills of arithmetic: adding, subtracting, multiplying and dividing whole numbers, fractions and decimals. If this goes on indefinitely, these students may eventually forget all or nearly all of these foundational skills.
— When students fail to learn (or stop recalling) fundamental math skills, they never learn — or forget —the underlying math concepts. For example, if students use calculators to add fractions with different denominators, and if eventually they always use a calculator for this operation, they will not grasp the many concepts that underlie this procedure: the concept of equivalent fractions, the skill of multiplying fractions, the techniques for finding the GCF and the LCM, and the list goes on.
— If students don’t grasp these critical underlying concepts and skills, their entire understanding of math will erode to the point of becoming “paper thin.” Mathematical knowledge is cumulative. So, for example, students who fail to understand fractions in general will never really understand how algebraic fractions work, nor fractions in trig formulas, nor fractions in the limit problems of calculus, or in any other area of math. These students will always be on a shaky footing whenever fractions pop up. And slipping around like that, conceptually, students will start to dislike math and subjects that rely on math, for it’s hard to like a subject if you don’t deeply understand it.
— Finally, calculator-reliant students either don’t develop — or lose — the ability to estimate, a critical and useful skill for daily life, for the rest of their lives. That is because estimating skills are dependent on the mental math skills of rounding off, adding, subtracting, multiplying, using percents, etc. When students become calculator-reliant, they often lose the ability to do these operations mentally. As a result, they become incapable of estimating spontaneously in all the situations of life that require estimation. (So if my car interest rate is 8%, how much will I pay per year in interest?)
Are you convinced by these arguments? Well, convinced or not, now we perform a “mental 180” to take a look at key arguments by those in favor of unrestricted calculator use:
— The calculator is a fixture of our modern high-tech world. In today’s workplace, skilled professionals (engineers, interior designers, financial planners, etc.) use calculators regularly, and there are no restrictions on their use there. So if today’s students hope to become tomorrow’s effective workers or entrepreneurs, they need to learn how to use calculators. It will help them understand math more deeply, and it will help them compete in the challenging economic conditions they’ll encounter.
— Calculators help students do math flawlessly. And after all, the point of math is to get the right answer. So if calculators help students get the right answer, students should be permitted to use calculators without restriction.
— Some students, due to learning disabilities, have great difficulty memorizing basic math facts, such as the facts of the times tables. These students should not be made to needlessly stress and suffer when calculators could help them access such facts quickly and accurately, whenever they need to do so.
— Most calculators today are used for more than just basic operations. Advanced calculators like graphing calculators (examples: the Texas Instruments Ti84 or NSpire , the Casio PRIZM models) feature a wide range of functions beyond simple calculations. As just a few examples, these calculators allow students to create graphs and charts, compute complex probabilities in real-life situations, perform complicated financial calculations, work with matrices, etc. To compete successfully for jobs, students will need to know how to use these sophisticated calculators. As home educators trying to prepare our children for their future, it’s incumbent that we help our kids learn how to use these indispensable tools.
Does it appear that there are good arguments on both sides? That’s what I think. But then, what do we, as home educators do?
First, I’d like to address a few comments on the arguments presented above. Then I’ll present my guidelines.
On the arguments against calculator use, a major point is that students who use calculators will become calculator-reliant, reaching for the calculators for virtually every calculation. I have seen this happen with students I tutor. I’ve seen 11th grade students grab the calculator for problems as simple as 7 + 5, or – 2 + 1, or 25% of 80.
So it does happen that students can become very calculator-reliant. But there are ways to introduce calculators so that this does not happen.
What about the point that students will forget basic facts, like the times tables? I’ve also seen students who once knew basic facts forget them. But again we, the parents, can control whether or not this happens, but how we introduce calculators.
I also tend to agree with the arguments that overuse of calculators can lead to students grasping little of the underlying principles, and that calculator-reliant students lose their skills for estimating. All of these are significant problems, but they can be avoided by careful planning by parents.
On the other side of the argument, there is validity in the argument that calculators are important tools in the workplace, and that we should teach students how to use them. Sometimes we homeschoolers are not aware of some of the general trends in public education, and I can say that the use of advanced calculators is definitely an important skill being taught, and for good reasons. To be educated for today’s workplace, students should know how to use advanced calculators.
The challenge is, how do we teach students how to use calculators without their ending up being calculator-reliant? That is what my 10 guidelines attempt to address. So here they are.
1) No calculator use from the beginning of a student’s math education through the end of the Grade 3 curriculum. During these key foundational years, we should require students to do all calculations using paper and pencil, or mentally, or best of all, using manipulatives in conjunction with paper-pencil techniques.
2) Starting in fourth grade, let students use calculators, but only to check their work. The idea is: students do their work. Then they check it with a calculator, just to see if it is right or wrong. They mark the problems that are wrong, and then the redo those problems. They can use a calculator to keep checking until they get the problem right. But — and this is important — require students to show all of their work. That way you can be assured that they are not using the calculator just to get the answer. If you are concerned that the child might be using the calculator to do the steps in a problem, just monitor to make sure that this is not happening.
3) General point. When you start to let students use calculators, make sure that you periodically quiz them on their basic facts: addition, subtraction, multiplication and division facts. You can find worksheets for these facts with a quick Google search. You don’t want your kids to lose these facts, and if you notice that this is happening, intervene, by all means!
4) In fifth grade, start to let students use calculators for higher purposes. One such higher purpose is to have students “play store” and add a complex sales tax to the cost of a purchase. Find your local sales tax, and teach the idea of sales tax to your child. Show your child how to add sales tax to a purchase by using a multiplier like 1.0625 for a tax of 6.25%. Then give your child a variety of purchase prices, and have your child figure out the cost after sales tax is added. You can take this a step further by having your child figure out the change to give you back if you pay with more than the exact amount.
5) Another higher purpose for calculators: help students discover patterns in multiplication. Example: ask your child what would be the highest product that can be made by multiplying a three-digit number by a two-digit number, with the restriction that only the digits 1, 2, 3, 4 and 5 may be used, and each number can be used once only. Using a calculator, your child can fairly quickly generate all of the possible products. After your child gets the answer — 431 x 52 = 22,412 is the highest product — ask why this would be the highest one. This can lead to an interesting discussion about place value.
6) Look for other resources on using calculators to explore math. One good one to get you started is Calculator Connections by Learning Resources.
7) Starting with 8th grade Algebra 1, get your child a graphing calculator and either teach them to use it, or find a tutor or program that will help them learn to use it.
8) Have your child keep using the advanced calculator for their work in all math at the level of Algebra 1 and beyond.
9) One other general point. A great bulwark against calculator-dependence is strong mental math skills. If you teach your children ways to do math computations quickly in their mind, they will be less tempted to use calculators, for they will have pride in their own ability to do mental math. There are many resources for mental math tricks, and there are even books on mental math tricks for children. One such is: Speed Math for Kids, by Bill Handley.
10) Finally, it helps to talk to your children about calculator use. Point out the importance of being able to do mental math in everyday life. Give them examples of how you use mental math in your life. Help your kids see that it’s in their own self-interest not to become calculator reliant. And ask them for their thoughts on calculator use. Keep the dialogue open.
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 book, Algebra Survival Guide, and the companion Algebra Survival Guide Workbook. Josh is also the author of the Card Game Roundup books, and has also written a playful approach to positive and negative numbers, PreAlgebra Blastoff!
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Check it out @ http://www.mathchat.wordpress.com Josh also leads workshops on math education for school districts large and small, and at homeschooling conferences. Josh can be reached at: josh@SingingTurtle.com
As I said in my last article, math often evokes strong feelings. So why not keep the juices flowing this month, too, by exploring a math issue that evokes passionate debate: the appropriate role of calculators in the homeschooling math curriculum.
Some people feel strongly that calculators have NO PLACE in math studies. On the opposite side, others believe students should have UNRESTRICTED ACCESS to calculators. So who’s right? To whom should you listen?
To me, of course, since I’m writing this article. (Haha!) But given the fact that you have no choice here and now, I’ll try to write this column in as open-minded a way as possible.
Whether you end up agreeing or disagreeing with my position, just thinking about this issue is a good thing, since it gives you a chance to develop or refine your ideas on the topic. For one thing, calculator usage is one of those issues you can’t avoid as a homeschooling parent, unless you live in a yurt and your kids have never even seen a calculator! Also, this is a decision with ramifications; it could very well affect how your children feel about math.
You may be happy to know that I am not an absolutist. That’s in part because, as a tutor for the past 20 or so years, I’ve seen so many different situations regarding kids and math and calculators that I’ve learned that there is no “one-size-fits-all” answer. But I have developed guidelines about calculator use, and that’s what I’ll lay out in this piece.
Before I present my own guidelines, though, let’s take a closer look at some of the arguments by people with strong views, for that gives us important points to consider.
First, some arguments by people who say calculators have ABSOLUTELY NO PLACE in the homeschooling math curriculum.
— Students permitted to use calculators will never memorize critical math facts, like addition or subtraction facts (12 – 5 = 7) or the times table facts (8 x 6 = 48). Or, if they once knew such facts, they will surely forget them, as soon as they become calculator-reliant.
— Calculators give students the message that they no longer have to rely on paper-pencil techniques, or on mental math abilities — to perform mathematical operations. By that I’m referring to operations like 24 x 17, 346 + 986, etc. Students can always “just pull out the calculator“ and start pushing buttons. And as students become more reliant on the calculator, they become weaker at fundamental math skills, such as the core skills of arithmetic: adding, subtracting, multiplying and dividing whole numbers, fractions and decimals. If this goes on indefinitely, these students may eventually forget all or nearly all of these foundational skills.
— When students fail to learn (or stop recalling) fundamental math skills, they never learn — or forget —the underlying math concepts. For example, if students use calculators to add fractions with different denominators, and if eventually they always use a calculator for this operation, they will not grasp the many concepts that underlie this procedure: the concept of equivalent fractions, the skill of multiplying fractions, the techniques for finding the GCF and the LCM, and the list goes on.
— If students don’t grasp these critical underlying concepts and skills, their entire understanding of math will erode to the point of becoming “paper thin.” Mathematical knowledge is cumulative. So, for example, students who fail to understand fractions in general will never really understand how algebraic fractions work, nor fractions in trig formulas, nor fractions in the limit problems of calculus, or in any other area of math. These students will always be on a shaky footing whenever fractions pop up. And slipping around like that, conceptually, students will start to dislike math and subjects that rely on math, for it’s hard to like a subject if you don’t deeply understand it.
— Finally, calculator-reliant students either don’t develop — or lose — the ability to estimate, a critical and useful skill for daily life, for the rest of their lives. That is because estimating skills are dependent on the mental math skills of rounding off, adding, subtracting, multiplying, using percents, etc. When students become calculator-reliant, they often lose the ability to do these operations mentally. As a result, they become incapable of estimating spontaneously in all the situations of life that require estimation. (So if my car interest rate is 8%, how much will I pay per year in interest?)
Are you convinced by these arguments? Well, convinced or not, now we perform a “mental 180” to take a look at key arguments by those in favor of unrestricted calculator use:
— The calculator is a fixture of our modern high-tech world. In today’s workplace, skilled professionals (engineers, interior designers, financial planners, etc.) use calculators regularly, and there are no restrictions on their use there. So if today’s students hope to become tomorrow’s effective workers or entrepreneurs, they need to learn how to use calculators. It will help them understand math more deeply, and it will help them compete in the challenging economic conditions they’ll encounter.
— Calculators help students do math flawlessly. And after all, the point of math is to get the right answer. So if calculators help students get the right answer, students should be permitted to use calculators without restriction.
— Some students, due to learning disabilities, have great difficulty memorizing basic math facts, such as the facts of the times tables. These students should not be made to needlessly stress and suffer when calculators could help them access such facts quickly and accurately, whenever they need to do so.
— Most calculators today are used for more than just basic operations. Advanced calculators like graphing calculators (examples: the Texas Instruments Ti84 or NSpire , the Casio PRIZM models) feature a wide range of functions beyond simple calculations. As just a few examples, these calculators allow students to create graphs and charts, compute complex probabilities in real-life situations, perform complicated financial calculations, work with matrices, etc. To compete successfully for jobs, students will need to know how to use these sophisticated calculators. As home educators trying to prepare our children for their future, it’s incumbent that we help our kids learn how to use these indispensable tools.
Does it appear that there are good arguments on both sides? That’s what I think. But then, what do we, as home educators do?
First, I’d like to address a few comments on the arguments presented above. Then I’ll present my guidelines.
On the arguments against calculator use, a major point is that students who use calculators will become calculator-reliant, reaching for the calculators for virtually every calculation. I have seen this happen with students I tutor. I’ve seen 11th grade students grab the calculator for problems as simple as 7 + 5, or – 2 + 1, or 25% of 80.
So it does happen that students can become very calculator-reliant. But there are ways to introduce calculators so that this does not happen.
What about the point that students will forget basic facts, like the times tables? I’ve also seen students who once knew basic facts forget them. But again we, the parents, can control whether or not this happens, but how we introduce calculators.
I also tend to agree with the arguments that overuse of calculators can lead to students grasping little of the underlying principles, and that calculator-reliant students lose their skills for estimating. All of these are significant problems, but they can be avoided by careful planning by parents.
On the other side of the argument, there is validity in the argument that calculators are important tools in the workplace, and that we should teach students how to use them. Sometimes we homeschoolers are not aware of some of the general trends in public education, and I can say that the use of advanced calculators is definitely an important skill being taught, and for good reasons. To be educated for today’s workplace, students should know how to use advanced calculators.
The challenge is, how do we teach students how to use calculators without their ending up being calculator-reliant? That is what my 10 guidelines attempt to address. So here they are.
1) No calculator use from the beginning of a student’s math education through the end of the Grade 3 curriculum. During these key foundational years, we should require students to do all calculations using paper and pencil, or mentally, or best of all, using manipulatives in conjunction with paper-pencil techniques.
2) Starting in fourth grade, let students use calculators, but only to check their work. The idea is: students do their work. Then they check it with a calculator, just to see if it is right or wrong. They mark the problems that are wrong, and then the redo those problems. They can use a calculator to keep checking until they get the problem right. But — and this is important — require students to show all of their work. That way you can be assured that they are not using the calculator just to get the answer. If you are concerned that the child might be using the calculator to do the steps in a problem, just monitor to make sure that this is not happening.
3) General point. When you start to let students use calculators, make sure that you periodically quiz them on their basic facts: addition, subtraction, multiplication and division facts. You can find worksheets for these facts with a quick Google search. You don’t want your kids to lose these facts, and if you notice that this is happening, intervene, by all means!
4) In fifth grade, start to let students use calculators for higher purposes. One such higher purpose is to have students “play store” and add a complex sales tax to the cost of a purchase. Find your local sales tax, and teach the idea of sales tax to your child. Show your child how to add sales tax to a purchase by using a multiplier like 1.0625 for a tax of 6.25%. Then give your child a variety of purchase prices, and have your child figure out the cost after sales tax is added. You can take this a step further by having your child figure out the change to give you back if you pay with more than the exact amount.
5) Another higher purpose for calculators: help students discover patterns in multiplication. Example: ask your child what would be the highest product that can be made by multiplying a three-digit number by a two-digit number, with the restriction that only the digits 1, 2, 3, 4 and 5 may be used, and each number can be used once only. Using a calculator, your child can fairly quickly generate all of the possible products. After your child gets the answer — 431 x 52 = 22,412 is the highest product — ask why this would be the highest one. This can lead to an interesting discussion about place value.
6) Look for other resources on using calculators to explore math. One good one to get you started is Calculator Connections by Learning Resources.
7) Starting with 8th grade Algebra 1, get your child a graphing calculator and either teach them to use it, or find a tutor or program that will help them learn to use it.
8) Have your child keep using the advanced calculator for their work in all math at the level of Algebra 1 and beyond.
9) One other general point. A great bulwark against calculator-dependence is strong mental math skills. If you teach your children ways to do math computations quickly in their mind, they will be less tempted to use calculators, for they will have pride in their own ability to do mental math. There are many resources for mental math tricks, and there are even books on mental math tricks for children. One such is: Speed Math for Kids, by Bill Handley.
10) Finally, it helps to talk to your children about calculator use. Point out the importance of being able to do mental math in everyday life. Give them examples of how you use mental math in your life. Help your kids see that it’s in their own self-interest not to become calculator reliant. And ask them for their thoughts on calculator use. Keep the dialogue open.
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 book, Algebra Survival Guide, and the companion Algebra Survival Guide Workbook. Josh is also the author of the Card Game Roundup books, and has also written a playful approach to positive and negative numbers, PreAlgebra Blastoff!
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Check it out @ http://www.mathchat.wordpress.com Josh also leads workshops on math education for school districts large and small, and at homeschooling conferences. Josh can be reached at: josh@SingingTurtle.com
